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Non-isomorphic abelian varieties with the same arithmetic.

Jamie Bell1

  • 1Department of Mathematics, University College London, London, UK.

Royal Society Open Science
|October 2, 2025
PubMed
Summary
This summary is machine-generated.

Researchers created two non-isomorphic abelian varieties over rational numbers. These varieties share isomorphic Mordell-Weil groups and Tate modules across all number fields, alongside other equal invariants.

Keywords:
abelianarithmeticnon-isomorphicvarieties

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Area of Science:

  • Number Theory
  • Algebraic Geometry

Background:

  • Abelian varieties are fundamental objects in algebraic geometry.
  • Understanding their properties, such as Mordell-Weil groups and Tate modules, is crucial for number theory research.

Purpose of the Study:

  • To construct and analyze abelian varieties with specific shared properties.
  • To investigate the relationship between isomorphism of abelian varieties and the isomorphism of their associated arithmetic invariants.

Main Methods:

  • Construction of two distinct abelian varieties over the field of rational numbers (ℚ).
  • Comparison of key arithmetic invariants, including Mordell-Weil groups and Tate modules, over various number fields.

Main Results:

  • The two constructed abelian varieties are not isomorphic.
  • Despite not being isomorphic, they exhibit isomorphic Mordell-Weil groups over every number field.
  • Isomorphic Tate modules and identical values for other significant invariants were also observed.

Conclusions:

  • Abelian varieties can share deep arithmetic properties like isomorphic Mordell-Weil groups and Tate modules even when they are not isomorphic themselves.
  • This finding highlights the complexity of classifying abelian varieties and suggests that certain arithmetic invariants may be more robust than the varieties themselves.